Rational Approximation Method for Stiff Initial Value Problems

While purely numerical methods for solving ordinary differential equations (ODE), e.g., Runge–Kutta methods, are easy to implement, solvers that utilize analytical derivations of the right-hand side of the ODE, such as the Taylor series method, outperform them in many cases. Nevertheless, the Taylor series method is not well-suited for stiff problems since it is explicit and not A-stable. In our paper, we present a numerical-analytical method based on the rational approximation of the ODE solution, which is naturally A- and A(α)-stable. We describe the rational approximation method and consider issues of order, stability, and adaptive step control. Finally, through examples, we prove the superior performance of the rational approximation method when solving highly stiff problems, comparing it with the Taylor series and Runge–Kutta methods of the same accuracy order.

The Taylor series method of order $p$ and Adams-Bashforth method on time scales

A recent study on the Taylor series method of second order and the trapezoidal rule for dynamic equations on time scales has been continued by introducing a derivation of the Taylor series method of arbitrary order $p$ on time scales. The error and convergence analysis of the method is also obtained. The 2 step Adams-Bashforth method for dynamic equations on time scales is concluded and applied to examples of initial value problems for nonlinear dynamic equations. Numerical results are presented and discussed.

The Double Phospho/Dephosphorylation Cycle as a Benchmark to Validate an Effective Taylor Series Method to Integrate Ordinary Differential Equations

The double phosphorylation/dephosphorylation cycle consists of a symmetric network of biochemical reactions of paramount importance in many intracellular mechanisms. From a network perspective, they consist of four enzymatic reactions interconnected in a specular way. The general approach to model enzymatic reactions in a deterministic fashion is by means of stiff Ordinary Differential Equations (ODEs) that are usually hard to integrate according to biologically meaningful parameter settings. Indeed, the quest for model simplification started more than one century ago with the seminal works by Michaelis and Menten, and their Quasi Steady-State Approximation methods are still matter of investigation nowadays. This work proposes an effective algorithm based on Taylor series methods that manages to overcome the problems arising in the integration of stiff ODEs, without settling for model approximations. The double phosphorylation/dephosphorylation cycle is exploited as a benchmark to validate the methodology from a numerical viewpoint.

NUMERICAL HYBRID ITERATIVE TECHNIQUE FOR SOLVING NONLINEAR EQUATIONS IN ONE VARIABLE

In recent years, some improvements have been suggested in the literature that has been a better performance or nearly equal to existing numerical iterative techniques (NIT). The efforts of this study are to constitute a Numerical Hybrid Iterative Technique (NHIT) for estimating the real root of nonlinear equations in one variable (NLEOV) that accelerates convergence. The goal of the development of the NHIT for the solution of an NLEOV assumed various efforts to combine the different methods. The proposed NHIT is developed by combining the Taylor Series method (TSM) and Newton Raphson’s iterative method (NRIM). MATLAB and Excel software has been used for the computational purpose. The developed algorithm has been tested on variant NLEOV problems and found the convergence is better than bracketing iterative method (BIM), which does not observe any pitfall and is almost equivalent to NRIM.

Approximating Soil Organic Carbon Stock in the Eastern Plains of Colombia

In Colombia, the rise of agricultural and pastureland expansion continues to exert increasing pressure on the structure and ecological processes of savannahs in the Eastern Plains. However, the effect of land use change on soil properties is often unknown due to poor access to remote areas. Effective management and conservation of soils requires the development spatial approaches that measure and predict dynamic soil properties such as soil organic carbon (SOC). This study estimates the SOC stock in the Eastern Plains of Colombia, with validation and uncertainty analyses, using legacy data of 653 soil samples. A random forest model of nine environmental covariate layers was used to develop predictions of SOC content. Model validation was determined using the Taylor series method, and root-mean-squared error (RMSE) and mean error (ME) were calculated to assess model performance. We found that the model explained 50.28% of the variation within digital SOC content map. Raster layers of SOC content, bulk density, and coarse rock fragment within the Eastern Plains were used to calculate SOC stock within the region. With uncertainty, SOC stock in the topsoil of the Eastern Plains was 1.2 G t ha−1. We found that SOC content contributed nearly all the uncertainty in the SOC stock predictions, although better determinations of SOC stock can be obtained with the use of a more geomorphological diverse dataset. The digital soil maps developed in this study provide predictions of extant SOC content and stock in the topsoil of the Eastern Plains, important soil information that may provide insight into the development of research, regulatory, and legislative initiatives to conserve and manage this evolving ecosystem.

Approximate Solution of Non-Linear Reaction-Diffusion in A Thin Membrane: Taylor Series Method

The nonlinear reaction-diffusion cycle in the thin membrane that describes the chemical reactions involving three species is studied. The model consists of the system of on nonlinear reaction-diffusion equations. The closed type of analytical expression of concentrations for the enzyme was developed by solving equations using the Taylor series formula. This results in the mixed Dirichlet and Neumann boundary conditions. Taylor series method similar to exponential function method. This technique provides approximate and simple solutions that are quick, easy to compute, and efficiently correct. These estimated findings are compared to the nuxmerical results. There is a good agreement with the simulation results.

Interval analysis-based Bi-iterative algorithm for robust TDOA-FDOA moving source localisation

In this study, an interval extension method of a bi-iterative is proposed to determine a moving source. This method is developed by utilising the time difference of arrival and frequency difference of arrival measurements of a signals received from several receivers. Unlike the standard Gaussian noise model, the time difference of arrival - frequency difference of arrival measurements are obtained by interval enclosing, which avoids convergence and initialisation problems in the conventional Taylor-series method. Using the bi-iterative strategy, the algorithm can alternately calculate the position and velocity of the moving source in interval vector form. Simulation results indicate that the proposed scheme significantly outperforms other methods, and approaches the Cramer-Rao lower bound at a sufficiently high noise level before the threshold effect occurs. Moreover, the interval widths of the results provide the confidence degree of the estimate.

Estimates in the Taylor series method for polynomial total systems of PDEs

Many of total systems of PDEs can be reduced to the polynomial form. As was shown by various authors, one of the best methods for the numerical solution of the initial value problem for ODE systems is the Taylor Series Method (TSM). In the article, the authors consider the Cauchy problem for the total polynomial PDE system, obtain the recurrence formulas for Taylor coefficients, and then formulate and prove a theorem on the accuracy of its solutions by TSM.